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Weak inhomogeneous Duffin–Schaeffer approximation

The shifted approximation problem fixes a real shift and asks for an almost-everywhere divergence criterion.

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Understanding the problem

Read the background

The shifted approximation problem fixes a real shift and asks for an almost-everywhere divergence criterion. Removing restrictions on that shift is separate from the homogeneous problem.

Numerators are unrestricted rather than required coprime. This is the weak inhomogeneous divergence statement, not a convergence converse or a quantitative counting asymptotic.

Why it matters

If correct, resolves the weak inhomogeneous divergence target for every fixed shift, including irrational shifts.

What counts as progress

A sourced resolution or substantial advance on the precise target, with hypotheses, prior context, and verification evidence stated separately.

What to watch for

  • Assess the precise statement and assumptions before comparing results.
  • Numerators are unrestricted rather than required coprime. This is the weak inhomogeneous divergence statement, not a convergence converse or a quantitative counting asymptotic.
  • Track independent assessments, dependencies, corrections, and withdrawals separately from importance.

Background review: 2026-10-11. Selected sources:

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